Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-5x+y &= -1 \\ 9x-9y &= 5\end{align*}$
Explanation: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = -9x+5$ Divide both sides by $-9$ to isolate $y$ $y = {x - \dfrac{5}{9}}$ Substitute this expression for $y$ in the first equation. $-5x+({x - \dfrac{5}{9}}) = -1$ $-5x + x - \dfrac{5}{9} = -1$ Simplify by combining terms, then solve for $x$ $-4x - \dfrac{5}{9} = -1$ $-4x = -\dfrac{4}{9}$ $x = \dfrac{1}{9}$ Substitute $\dfrac{1}{9}$ for $x$ back into the top equation. $-5( \dfrac{1}{9})+y = -1$ $-\dfrac{5}{9}+y = -1$ $y = -\dfrac{4}{9}$ $y = -\dfrac{4}{9}$ The solution is $\enspace x = \dfrac{1}{9}, \enspace y = -\dfrac{4}{9}$.